Problem: $ C = \left[\begin{array}{rr}2 & -1 \\ 4 & 4 \\ 2 & 2\end{array}\right]$ $ B = \left[\begin{array}{rr}0 & 1 \\ -2 & 0\end{array}\right]$ What is $ C B$ ?
Solution: Because $ C$ has dimensions $(3\times2)$ and $ B$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C B = \left[\begin{array}{rr}{2} & {-1} \\ {4} & {4} \\ \color{gray}{2} & \color{gray}{2}\end{array}\right] \left[\begin{array}{rr}{0} & \color{#DF0030}{1} \\ {-2} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{0}+{-1}\cdot{-2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{0}+{-1}\cdot{-2} & ? \\ {4}\cdot{0}+{4}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{0}+{-1}\cdot{-2} & {2}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{0} \\ {4}\cdot{0}+{4}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{0}+{-1}\cdot{-2} & {2}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{0} \\ {4}\cdot{0}+{4}\cdot{-2} & {4}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{0} \\ \color{gray}{2}\cdot{0}+\color{gray}{2}\cdot{-2} & \color{gray}{2}\cdot\color{#DF0030}{1}+\color{gray}{2}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}2 & 2 \\ -8 & 4 \\ -4 & 2\end{array}\right] $